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252 Psychosocial factors influencing the maintenance of a healthy lifestyle among African American adults during the COVID-19 Pandemic
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- Carrie R. McCoy, Ashya Burgess, Clarence Jones, Monisha Richard, Minnesota Jamia Erickson, Irene G. Sia, Mark L. Wieland, Chyke A. Doubeni, LaPrincess C. Brewer
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- Journal:
- Journal of Clinical and Translational Science / Volume 8 / Issue s1 / April 2024
- Published online by Cambridge University Press:
- 03 April 2024, p. 76
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OBJECTIVES/GOALS: The COVID-19 pandemic disrupted established social support networks (faith-based, community, family, friends), resulting in unprecedented health-related, financial, and employment challenges among African Americans (AAs). This study explores the psychosocial influences of the pandemic on the health and wellness of AAs. METHODS/STUDY POPULATION: The FAITH! (Fostering African-American Improvement in Total Health!) Program, an academic-community partnership with AA churches, shifted focus to COVID-19 prevention in AA communities. Funded by the Mayo Clinic Center for Clinical and Translation Sciences, this cross-sectional study recruited AA adults from FAITH!-affiliated churches and social media to complete a survey exploring the personal impact of the pandemic from hardships (e.g., food and housing insecurity, paying utilities) on healthy lifestyle (HL). The primary outcome was difficulty maintaining a HL during the pandemic. Logistic regression (odds ratios and associated 95% confidence intervals (CIs)) was used to examine the associations between difficulty maintaining a HL and factors including COVID-19 hardships and mental health. RESULTS/ANTICIPATED RESULTS: Participants (N=169, 71.4% female, 41.4% essential workers) had a mean age [SD] of 49.4 [14.9] years. Over half (91/169, 54%) reported difficulty maintaining a HL. Those reporting unemployment (OR 2.3; 95% CI [1.2,4.4]; p=0.008), difficulty paying rent (OR 4.1; 95% CI [2.1,8.6]; p<0.001), or food/utilities (OR 5.5; 95% CI [2.7,11.5]; p<0.001) all had greater odds of difficulty maintaining a HL. High stress (≥5/10, scale 1-10) was associated with difficulty maintaining a HL (OR 4.1; 95% CI [2.1,8.5]; p<0.001) compared to AAs with low stress. Negative mental health (depression (OR 3.4; 95% CI [1.0,13.7]; p<0.001), anger (OR 2.5; 95% CI [0.5,18.9]; p=0.005), and nervousness (OR 4.1; 95% CI [1.1,19.5]; p=0.003) was associated with difficulty maintaining a HL compared to AAs with positive mental health. DISCUSSION/SIGNIFICANCE: Our study findings revealed that COVID-19 hardships, stress, and negative mental health impacted the ability of AAs to maintain a HL. These issues should be considered in the design and implementation of community-based health programs to promote healthy living during future public health emergencies.
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- By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. Dillon, Robert DiSalle, Mary Domski, Alan Donagan, Paul Draper, Fred Dretske, Mircea Dumitru, Wilhelm Dupré, Gerald Dworkin, John Earman, Ellery Eells, Catherine Z. Elgin, Berent Enç, Ronald P. Endicott, Edward Erwin, John Etchemendy, C. Stephen Evans, Susan L. Feagin, Solomon Feferman, Richard Feldman, Arthur Fine, Maurice A. Finocchiaro, William FitzPatrick, Richard E. Flathman, Gvozden Flego, Richard Foley, Graeme Forbes, Rainer Forst, Malcolm R. Forster, Daniel Fouke, Patrick Francken, Samuel Freeman, Elizabeth Fricker, Miranda Fricker, Michael Friedman, Michael Fuerstein, Richard A. Fumerton, Alan Gabbey, Pieranna Garavaso, Daniel Garber, Jorge L. A. Garcia, Robert K. Garcia, Don Garrett, Philip Gasper, Gerald Gaus, Berys Gaut, Bernard Gert, Roger F. Gibson, Cody Gilmore, Carl Ginet, Alan H. Goldman, Alvin I. Goldman, Alfonso Gömez-Lobo, Lenn E. Goodman, Robert M. Gordon, Stefan Gosepath, Jorge J. E. Gracia, Daniel W. Graham, George A. Graham, Peter J. Graham, Richard E. Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. Lemos, Ernest LePore, Sarah-Jane Leslie, Isaac Levi, Andrew Levine, Alan E. Lewis, Daniel E. Little, Shu-hsien Liu, Shu-hsien Liu, Alan K. L. Chan, Brian Loar, Lawrence B. Lombard, John Longeway, Dominic McIver Lopes, Michael J. Loux, E. J. Lowe, Steven Luper, Eugene C. Luschei, William G. Lycan, David Lyons, David Macarthur, Danielle Macbeth, Scott MacDonald, Jacob L. Mackey, Louis H. Mackey, Penelope Mackie, Edward H. Madden, Penelope Maddy, G. B. Madison, Bernd Magnus, Pekka Mäkelä, Rudolf A. Makkreel, David Manley, William E. Mann (W.E.M.), Vladimir Marchenkov, Peter Markie, Jean-Pierre Marquis, Ausonio Marras, Mike W. Martin, A. P. Martinich, William L. McBride, David McCabe, Storrs McCall, Hugh J. McCann, Robert N. McCauley, John J. McDermott, Sarah McGrath, Ralph McInerny, Daniel J. McKaughan, Thomas McKay, Michael McKinsey, Brian P. McLaughlin, Ernan McMullin, Anthonie Meijers, Jack W. Meiland, William Jason Melanson, Alfred R. Mele, Joseph R. Mendola, Christopher Menzel, Michael J. Meyer, Christian B. Miller, David W. Miller, Peter Millican, Robert N. Minor, Phillip Mitsis, James A. Montmarquet, Michael S. Moore, Tim Moore, Benjamin Morison, Donald R. Morrison, Stephen J. Morse, Paul K. Moser, Alexander P. D. Mourelatos, Ian Mueller, James Bernard Murphy, Mark C. Murphy, Steven Nadler, Jan Narveson, Alan Nelson, Jerome Neu, Samuel Newlands, Kai Nielsen, Ilkka Niiniluoto, Carlos G. Noreña, Calvin G. Normore, David Fate Norton, Nikolaj Nottelmann, Donald Nute, David S. Oderberg, Steve Odin, Michael O’Rourke, Willard G. Oxtoby, Heinz Paetzold, George S. Pappas, Anthony J. Parel, Lydia Patton, R. P. Peerenboom, Francis Jeffry Pelletier, Adriaan T. Peperzak, Derk Pereboom, Jaroslav Peregrin, Glen Pettigrove, Philip Pettit, Edmund L. Pincoffs, Andrew Pinsent, Robert B. Pippin, Alvin Plantinga, Louis P. Pojman, Richard H. Popkin, John F. Post, Carl J. Posy, William J. Prior, Richard Purtill, Michael Quante, Philip L. Quinn, Philip L. Quinn, Elizabeth S. Radcliffe, Diana Raffman, Gerard Raulet, Stephen L. Read, Andrews Reath, Andrew Reisner, Nicholas Rescher, Henry S. Richardson, Robert C. Richardson, Thomas Ricketts, Wayne D. Riggs, Mark Roberts, Robert C. Roberts, Luke Robinson, Alexander Rosenberg, Gary Rosenkranz, Bernice Glatzer Rosenthal, Adina L. Roskies, William L. Rowe, T. M. Rudavsky, Michael Ruse, Bruce Russell, Lilly-Marlene Russow, Dan Ryder, R. M. Sainsbury, Joseph Salerno, Nathan Salmon, Wesley C. Salmon, Constantine Sandis, David H. Sanford, Marco Santambrogio, David Sapire, Ruth A. Saunders, Geoffrey Sayre-McCord, Charles Sayward, James P. Scanlan, Richard Schacht, Tamar Schapiro, Frederick F. Schmitt, Jerome B. Schneewind, Calvin O. Schrag, Alan D. Schrift, George F. Schumm, Jean-Loup Seban, David N. Sedley, Kenneth Seeskin, Krister Segerberg, Charlene Haddock Seigfried, Dennis M. Senchuk, James F. Sennett, William Lad Sessions, Stewart Shapiro, Tommie Shelby, Donald W. Sherburne, Christopher Shields, Roger A. Shiner, Sydney Shoemaker, Robert K. Shope, Kwong-loi Shun, Wilfried Sieg, A. John Simmons, Robert L. Simon, Marcus G. Singer, Georgette Sinkler, Walter Sinnott-Armstrong, Matti T. Sintonen, Lawrence Sklar, Brian Skyrms, Robert C. Sleigh, Michael Anthony Slote, Hans Sluga, Barry Smith, Michael Smith, Robin Smith, Robert Sokolowski, Robert C. Solomon, Marta Soniewicka, Philip Soper, Ernest Sosa, Nicholas Southwood, Paul Vincent Spade, T. L. S. Sprigge, Eric O. Springsted, George J. Stack, Rebecca Stangl, Jason Stanley, Florian Steinberger, Sören Stenlund, Christopher Stephens, James P. Sterba, Josef Stern, Matthias Steup, M. A. Stewart, Leopold Stubenberg, Edith Dudley Sulla, Frederick Suppe, Jere Paul Surber, David George Sussman, Sigrún Svavarsdóttir, Zeno G. Swijtink, Richard Swinburne, Charles C. Taliaferro, Robert B. Talisse, John Tasioulas, Paul Teller, Larry S. Temkin, Mark Textor, H. S. Thayer, Peter Thielke, Alan Thomas, Amie L. Thomasson, Katherine Thomson-Jones, Joshua C. Thurow, Vzalerie Tiberius, Terrence N. Tice, Paul Tidman, Mark C. Timmons, William Tolhurst, James E. Tomberlin, Rosemarie Tong, Lawrence Torcello, Kelly Trogdon, J. D. Trout, Robert E. Tully, Raimo Tuomela, John Turri, Martin M. Tweedale, Thomas Uebel, Jennifer Uleman, James Van Cleve, Harry van der Linden, Peter van Inwagen, Bryan W. Van Norden, René van Woudenberg, Donald Phillip Verene, Samantha Vice, Thomas Vinci, Donald Wayne Viney, Barbara Von Eckardt, Peter B. M. Vranas, Steven J. Wagner, William J. Wainwright, Paul E. Walker, Robert E. Wall, Craig Walton, Douglas Walton, Eric Watkins, Richard A. Watson, Michael V. Wedin, Rudolph H. Weingartner, Paul Weirich, Paul J. Weithman, Carl Wellman, Howard Wettstein, Samuel C. Wheeler, Stephen A. White, Jennifer Whiting, Edward R. Wierenga, Michael Williams, Fred Wilson, W. Kent Wilson, Kenneth P. Winkler, John F. Wippel, Jan Woleński, Allan B. Wolter, Nicholas P. Wolterstorff, Rega Wood, W. Jay Wood, Paul Woodruff, Alison Wylie, Gideon Yaffe, Takashi Yagisawa, Yutaka Yamamoto, Keith E. Yandell, Xiaomei Yang, Dean Zimmerman, Günter Zoller, Catherine Zuckert, Michael Zuckert, Jack A. Zupko (J.A.Z.)
- Edited by Robert Audi, University of Notre Dame, Indiana
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- The Cambridge Dictionary of Philosophy
- Published online:
- 05 August 2015
- Print publication:
- 27 April 2015, pp ix-xxx
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. Douglas Meeks, Monica Jyotsna Melanchthon, Ilie Melniciuc-Puica, Everett Mendoza, Raymond A. Mentzer, William W. Menzies, Ina Merdjanova, Franziska Metzger, Constant J. Mews, Marvin Meyer, Carol Meyers, Vasile Mihoc, Gunner Bjerg Mikkelsen, Maria Inêz de Castro Millen, Clyde Lee Miller, Bonnie J. Miller-McLemore, Alexander Mirkovic, Paul Misner, Nozomu Miyahira, R. W. L. Moberly, Gerald Moede, Aloo Osotsi Mojola, Sunanda Mongia, Rebeca Montemayor, James Moore, Roger E. Moore, Craig E. Morrison O.Carm, Jeffry H. Morrison, Keith Morrison, Wilson J. Moses, Tefetso Henry Mothibe, Mokgethi Motlhabi, Fulata Moyo, Henry Mugabe, Jesse Ndwiga Kanyua Mugambi, Peggy Mulambya-Kabonde, Robert Bruce Mullin, Pamela Mullins Reaves, Saskia Murk Jansen, Heleen L. Murre-Van den Berg, Augustine Musopole, Isaac M. T. Mwase, Philomena Mwaura, Cecilia Nahnfeldt, Anne Nasimiyu Wasike, Carmiña Navia Velasco, Thulani Ndlazi, Alexander Negrov, James B. Nelson, David G. Newcombe, Carol Newsom, Helen J. Nicholson, George W. E. Nickelsburg, Tatyana Nikolskaya, Damayanthi M. A. Niles, Bertil Nilsson, Nyambura Njoroge, Fidelis Nkomazana, Mary Beth Norton, Christian Nottmeier, Sonene Nyawo, Anthère Nzabatsinda, Edward T. Oakes, Gerald O'Collins, Daniel O'Connell, David W. Odell-Scott, Mercy Amba Oduyoye, Kathleen O'Grady, Oyeronke Olajubu, Thomas O'Loughlin, Dennis T. Olson, J. Steven O'Malley, Cephas N. Omenyo, Muriel Orevillo-Montenegro, César Augusto Ornellas Ramos, Agbonkhianmeghe E. Orobator, Kenan B. Osborne, Carolyn Osiek, Javier Otaola Montagne, Douglas F. Ottati, Anna May Say Pa, Irina Paert, Jerry G. Pankhurst, Aristotle Papanikolaou, Samuele F. Pardini, Stefano Parenti, Peter Paris, Sung Bae Park, Cristián G. Parker, Raquel Pastor, Joseph Pathrapankal, Daniel Patte, W. Brown Patterson, Clive Pearson, Keith F. Pecklers, Nancy Cardoso Pereira, David Horace Perkins, Pheme Perkins, Edward N. Peters, Rebecca Todd Peters, Bishop Yeznik Petrossian, Raymond Pfister, Peter C. Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
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21 - Monadic and Dyadic Logic
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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Summary
We have given in earlier chapters several different proofs of Church's theorem to the effect that first-order logic is undecidable: there is no effective procedure that applied to any first-order sentence will in a finite amount of time tell us whether or not it is valid. This negative result leaves room on the one hand for contrasting positive results, and on the other hand for sharper negative results. The most striking of the former is the Löwenheim–Behmann theorem, to the effect that the logic of monadic (one-place) predicates is decidable, even when the two-place logical predicate of identity is admitted. The most striking of the latter is the Church–Herbrand theorem that the logic of a single dyadic (two-place) predicate is undecidable. These theorems are presented in sections 21.2 and 21.3 after some general discussion of solvable and unsolvable cases of the decision problem for logic in section 21.1. While the proof of Church's theorem requires the use of considerable computability theory (the theory of recursive functions, or of Turing machines), that is not so for the proof of the Löwenheim–Behmann theorem or for the proof that Church's theorem implies the Church–Herbrand theorem. The former uses only material developed by Chapter 11. The latter uses also the elimination of function symbols and identity from section 19.4, but nothing more than this. The proofs of these two results, positive and negative, are independent of each other.
Contents
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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BASIC METALOGIC
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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COMPUTABILITY THEORY
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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9 - A Précis of First-Order Logic: Syntax
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Summary
This chapter and the next contain a summary of material, mainly definitions, needed for later chapters, of a kind that can be found expounded more fully and at a more relaxed pace in introductory-level logic textbooks. Section 9.1 gives an overview of the two groups of notions from logical theory that will be of most concern: notions pertaining to formulas and sentences, and notions pertaining to truth under an interpretation. The former group of notions, called syntactic, will be further studied in section 9.2, and the latter group, called semantic, in the next chapter.
First-Order Logic
Logic has traditionally been concerned with relations among statements, and with properties of statements, that hold by virtue of ‘form’ alone, regardless of ‘content’. For instance, consider the following argument:
(1) A mother or father of a person is an ancestor of that person.
(2) An ancestor of an ancestor of a person is an ancestor of that person.
(3) Sarah is the mother of Isaac, and Isaac is the father of Jacob.
(4) Therefore, Sarah is an ancestor of Jacob.
Logic teaches that the premisses (1)–(3) (logically) imply or have as a (logical) consequence the conclusion (4), because in any argument of the same form, if the premisses are true, then the conclusion is true.
10 - A Précis of First-Order Logic: Semantics
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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8 - Equivalent Definitions of Computability
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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In the preceding several chapters we have introduced the intuitive notion of effective computability, and studied three rigorously defined technical notions of computability: Turing computability, abacus computability, and recursive computability, noting along the way that any function that is computable in any of these technical senses is computable in the intuitive sense. We have also proved that all recursive functions are abacus computable and that all abacus-computable functions are Turing computable. In this chapter we close the circle by showing that all Turing-computable functions are recursive, so that all three notions of computability are equivalent. It immediately follows that Turing's thesis, claiming that all effectively computable functions are Turing computable, is equivalent to Church's thesis, claiming that all effectively computable functions are recursive. The equivalence of these two theses, originally advanced independently of each other, does not amount to a rigorous proof of either, but is surely important evidence in favor of both. The proof of the recursiveness of Turing-computable functions occupies section 8.1. Some consequences of the proof of equivalence of the three notions of computability are pointed out in section 8.2, the most important being the existence of a universal Turing machine, a Turing machine capable of simulating the behavior of any other Turing machine desired. The optional section 8.3 rounds out the theory of computability by collecting basic facts about recursively enumerable sets, sets of natural numbers that can be enumerated by a recursive function. […]
7 - Recursive Sets and Relations
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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In the preceding chapter we introduced the classes of primitive recursive and recursive functions. In this chapter we introduce the related notions of primitive recursive and recursive sets and relations, which help provide many more examples of primitive recursive and recursive functions. The basic notions are developed in section 7.1. Section 7.2 introduces the related notion of a semirecursive set or relation. The optional section 7.3 presents examples of recursive total functions that are not primitive recursive.
Recursive Relations
A set of, say, natural numbers is effectively decidable if there is an effective procedure that, applied to a natural number, in a finite amount of time gives the correct answer to the question whether it belongs to the set. Thus, representing the answer ‘yes’ by 1 and the answer ‘no’ by 0, a set is effectively decidable if and only if its characteristic function is effectively computable, where the characteristic function is the function that takes the value 1 for numbers in the set, and the value 0 for numbers not in the set. A set is called recursively decidable, or simply recursive for short, if its characteristic function is recursive, and is called primitive recursive if its characteristic function is primitive recursive. Since recursive functions are effectively computable, recursive sets are effectively decidable. Church's thesis, according to which all effectively computable functions are recursive, implies that all effectively decidable sets are recursive.
18 - The Unprovability of Consistency
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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FURTHER TOPICS
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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24 - Decidability of Arithmetic without Multiplication
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26 - Ramsey's Theorem
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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Summary
Ramsey's theorem is a combinatorial result about finite sets with a proof that has interesting logical features. To prove this result about finite sets, we are first going to prove, in section 26.1, an analogous result about infinite sets, and are then going to derive, in section 26.2, the finite result from the infinite result. The derivation will be an application of the compactness theorem. Nothing in the proof of Ramsey's theorem to be presented requires familiarity with logic beyond the statement of the compactness theorem, but at the end of the chapter we indicate how Ramsey theory provides an example of a sentence undecidable in P that is more natural mathematically than any we have encountered so far.
Ramsey's Theorem: Finitary and Infinitary
There is an old puzzle about a party attended by six persons, at which any two of the six either like each other or dislike each other: the problem is to show that at the party there are three persons, any two of whom like each other, or there are three persons, any two of whom dislike each other.
The solution: Let a be one of the six. Since there are five others, either there will be (at least) three others that a likes or there will be three others that a dislikes. Suppose a likes them. (The argument is similar if a dislikes them.) Call the three b, c, d.
11 - The Undecidability of First-Order Logic
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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This chapter connects our work on computability with questions of logic. Section 11.1 presupposes familiarity with the notions of logic from Chapter 9 and 10 and of Turing computability from Chapters 3–4, including the fact that the halting problem is not solvable by any Turing machine, and describes an effective procedure for producing, given any Turing machine M and input n, a set of sentences Г and a sentence D such that M given input n will eventually halt if and only if Г implies D. It follows that if there were an effective procedure for deciding when a finite set of sentences implies another sentence, then the halting problem would be solvable; whereas, by Turing's thesis, the latter problem is not solvable, since it is not solvable by a Turing machine. The upshot is, one gets an argument, based on Turing's thesis for (the Turing–Büchi proof of) Church's theorem, that the decision problem for implication is not effectively solvable. Section 11.2 presents a similar argument–the Gödel-style proof of Church's theorem–this time using not Turing machines and Turing's thesis, but primitive recursive and recursive functions and Church's thesis, as in Chapters 6–7. The constructions of the two sections, which are independent of each other, are both instructive; but an entirely different proof, not dependent on Turing's or Church's thesis, will be given in a later chapter, and in that sense the present chapter is optional. […]
12 - Models
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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20 - The Craig Interpolation Theorem
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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Suppose that a sentence A implies a sentence C. The Craig interpolation theorem tells us that in that case there is a sentence B such that A implies B, B implies C, and B involves no nonlogical symbols but such as occur both in A and in B. This is one of the basic results of the theory of models, almost on a par with, say, the compactness theorem. The proof is presented in section 20.1. The proof for the special case where identity and function symbols are absent is an easy further application of the same lemmas that we have applied to prove the compactness theorem in Chapter 13, and could have been presented there. But the easiest proof for the general case is by reduction to this special case, using the machinery for the elimination of function symbols and identity developed in section 19.4. Sections 20.2 and 20.3, which are independent of each other, take up two significant corollaries of the interpolation theorem, Robinson's joint consistency theorem and Beth's definability theorem.
Craig's Theorem and Its Proof
We begin with a simple observation.
Proposition. If a sentence A implies a sentence C, then there is a sentence B that A implies, that implies C, and that contains only such constants as are contained in both of A and C.
23 - Arithmetical Definability
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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Tarski's theorem tells us that the set V of (code numbers of) first-order sentences of the language in arithmetic that are true in the standard interpretation is not arithmetically definable. In section 23.1 we show that this negative result is poised, so to speak, between two positive results. One is that for each n the set Vn of sentences of the language of arithmetic of degree of complexity n that are true in the standard interpretation is arithmetically definable (in a sense of degree of complexity to be made precise). The other is that the class {V} of sets of natural numbers whose one and only member is V is arithmetically definable (in a sense of arithmetical definability for classes to be made precise). In section 23.2 we take up the question whether the class of arithmetically definable sets of numbers is an arithmetically definable class of sets. The answer is negative, according to Addison's theorem. This result is perhaps most interesting on account of its method of proof, which is a comparatively simple application of the method of forcing originally devised to prove the independence of the continuum hypothesis in set theory (as alluded to in the historical notes to Chapter 18).
Arithmetical Definability and Truth
Throughout this chapter we use L and N for the language of arithmetic and its standard interpretation (previously called L* and N*), and V for the set of code numbers of first-order sentences of L ture in N.
Index
- George S. Boolos, John P. Burgess, Princeton University, New Jersey, Richard C. Jeffrey
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